When counting only for the sake of counting, i.e. without any actual things being counted (like in a game such as hide and seek), which gender would be most idiomatic? Similarly when demonstrating arithmetics. Am I right to assume that the neuter should be used, i.e. "unum duo tria quattuor...", "unum et tria faciunt quattuor", etc.?

Thanks Adrianus! However, the ubiquitous example of multiplication in (modern) grammars seems to be "bis bina sunt quattuor", rather than "bini". Is there any special reason for this, or is it simply the case that it doesn't matter much?

For "unus duo tres", the Greek's say "unum duo tria" (that is, the Greeks use the neuter form in counting). That means that, if you were taught arithmetic by a Greek servant, you used neuter gender for numbers, but masculine if taught otherwise. Interesting. Following the methods of Comenius, I'm trying to create a computer-interactive schoolroom for the immersive learning of Latin but with a twist, so this is all useful to me.

I thought I understood things, but your question topic now has me not 100% uncertain, although I think Isadore is being clear on counting and calculating in the masculine gender (unless the context dictates otherwise, with feminine "pars", say, or neuter "mille, millia"). On the subject of numbers, this subject has the potential to annoy some people. I have an interactive playing-card puzzle that I showed at a Latin conference two years ago. It made one teacher rather irritated because, instead of "unus duo tres..." I had programmed the cards to be spoken as "monas, dyas, ternio, quaternio, pentas, senio, heptas, ogdoas, enneas, decas" (all of which are singular nouns, not pluralâ€”meaning "[una] decas" = "a ten", say, or "duae decades" = "two tens") because that's how people played cards in Latin, I believe, until the nineteenth century. We still do in English, of course, use "ace" and "deuce", but "terce" (for 3) etc is no longer used.

Nothing, therefore, could be more ignorant than to assert, as you tend to do, that one cannot say for certain that the very world itself is round, for it could be another shape,â€”there being infinite other [possible] worlds of other shapes. Which is something that Epicurus, if he had learned to put two and two together, would certainly never have said; rather, whilst the roof of his own mouth guided his taste or judgement, he did not look to the "roof of heaven", as Ennius puts it.